A new evaluation method of a block error probability of a block modulation code over an AWGN channel

Let L be a finite set of symbols, and let C be a block code of length n over L. Let h be a positive integer. For u∈L, let s(u) denote the signal point in ℛ^h which represents u, where ℛ^h denotes the set of all h-tuples of real numbers, and for u=(u₁,u₂,…,u_n) over L, let s(u) denote the hn-tuple (s(u₁),s(u₂),…,s(u _n)). For z and z´ in ℛ^hn let ||z-z´|| denote the Euclidean distance between z and z´. For u∈C/{0}, let pe(r,u) denote the probability of an incorrect decoding of s(0) into u under the condition that ||s(0)-z||=r where r is a non-negative real number. Then, f(r)=Σ_u∈C{0}/pe(r,u). For u∈C{0}, let C_u be a subcode of C which contains u. Let P(r,C_u) denote the probability that for every v∈C_u other than u, a received z satisfies ||s(u)-z||⩽||s(v)-z|| under the condition that ||s(0)-z||=r. In the examples given, we present a tighter evaluation method of pe(r,u) or the sum of pe(r,u) over all the nearest neighbor us of 0